The Bungee Jumper: A
Comparison of Predicted
and Measured Values
Hubert Biezeveld, OSG West-Friesland, Hoorn, The Netherlands
T
he greater-than-g acceleration of a bungee
jumper discussed in a previous article in this
journal by Kagan and Kott1 led to many lively discussions among Dutch physics teachers. These
inspired me to look for an inexpensive experimental
setup, suitable for use in a high school physics class,
that can be used to confirm that indeed the acceleration is greater than g. In this paper I describe an exercise to compare the predicted and the measured
graphs for the displacement y(t) of the jumper and the
force Fb(t) exerted by the bungee on the bridge to
which it is fastened. In my apparatus, the “bungee”
consists of a light chain and the “jumper” is a small
piece of brass. Data collection and the calculation of
predicted values were carried out using Coach.2 The
analysis reliably leads to the conclusion that the acceleration of the falling jumper does indeed exceed g.
Theory
A real bungee jumper (a person of mass M) is tied
to an elastic cord (bungee) of mass m and length L.
The other end of the rope is tied to a fixed point —
perhaps on a bridge. Starting from rest on the bridge,
the jumper accelerates downward to speed v after
falling a distance y. Using conservation of mechanical
energy, Kagan and Kott show that while the cord is
still slack:
(4ML + 2mL – my)
v2 = gy .
(mL – my + 2mL)
And differentiating:
238
(1)
my(4ML+2mL–my)
a = g 1+ .
2(mL–my+2ML)2
冤
冥
(2)
A solution for y(t) may be obtained in terms of elliptic integrals.3 This is inappropriate for the introductory-level physics course, however. Numerical integration using, for example, a spreadsheet is certainly
preferable. Commenting about the counterintuitive
conclusion that a > g, Kagan and Kott write: “Some
insight can be gleaned from thinking about the fact
that the portion of cord at the bottom of the loop is
coming to rest. Since it was moving downward, an
upward force is required on this part of the system.”
This remark put me on the track to find a simple
algorithm for determining numerically the displacement y(t) of the jumper and also the force Fb(t)
exerted by the bungee on the bridge. My analysis is
limited to the time during which the bungee remains
slack.
Algorithm
The method consists of N successive loops in which
we continually calculate new values for t, y, and Fb. We
start with the values of the various constants employed
in our experimental setup. The length and mass of the
chain (bungee) are L = 1.51 m and m = 0.290 kg. The
mass of the brass cylinder (jumper) is 0.057 kg. The
linear density of the bungee is = m/L. We divide the
length L into N (N = 10,000 for the calculations described here) pieces of length dy = L/N, each having
mass dy. After the values of the constants have been
set, the iterative process may be initiated.
DOI: 10.1119/1.1564507
THE PHYSICS TEACHER ◆ Vol. 41, April 2003
Fig. 2. A brass chain is used for the bungee.
•
Fig. 1(a). Bungee jumper system at time t.
Fig. 1(b). The system
at time t + dt.
•
The following is an annotated list of the steps contained in each of the N calculation loops:
•
•
•
•
•
•
At the start of a loop, the elapsed time is t, and the
jumper has fallen a distance y. Referring to Fig.
1(a), we see that a length ½(L + y) of the bungee
(on the left side) is at rest. On the right side we
have both the jumper and a length ½(L – y) of
bungee moving with velocity v1 [see Eq. (1)].
We add dy to y and write y := y + dy (:= stands for
“the old value is replaced by ...”). From this new y
we calculate a new velocity v2 using Eq. (1).
The mean velocity vm = (v1 + v2)/2 is calculated.
The time dt needed for the displacement dy is dt =
dy/vm. This value of dt is added to t, and t := t + dt.
When the jumper descends the distance dy a portion ½dy of the bungee having mass ½dy and velocity v1 at the right side of the bottom of the loop
[Fig. 1(b)] comes to rest on the left side.
To bring this mass to rest, a force F = (½dy)v1/dt
(Newton’s second law) is required. Since the tension in the bungee is continuous,3 half of this force
THE PHYSICS TEACHER ◆ Vol. 41, April 2003
is supplied by the left side and the other half by the
right side of the bungee.
The total force Fb on the bridge may be written as:
Fb = ½(L + y)g + ½ (½dy)v1/dt.
The first term is the static weight of the left side of
the bungee; the second term is half of the dynamic
force just calculated. This force Fb can be measured with a force sensor.
To start a new loop we simply replace v1 by v2, i.e.,
v1 := v2.
Measurements and Comparison to
Predicted Values
As mentioned earlier, in this experiment the
jumper is made of brass. It is actually connected to
two chains (bungees) (see Fig. 2) in order to reduce
the possibility of horizontal motion during the fall.
The jumper is constructed so that it can slide down a
constantan wire across which a voltage of 5.00 V is applied. This setup constitutes a potentiometer that can
be calibrated as a y-sensor. See Appendix for a detailed
description.
Figure 3 shows a comparison of my measured y(t)
for the jumper (red) and y(t) for free fall, i.e., y = 4.9t 2
(green). It is evident that the bungee jumper does indeed have a greater-than-g acceleration. Figure 4
shows the measured y(t) (red) and y(t) as determined
using Eq. (1) and the algorithm described (green).
The agreement is very good.
The force Fb exerted by the fixed support on one of
the two chains was measured with a standard force
239
Fig. 3. Experimental y(t) graph for the jumper (red) and
for a mass in free fall [y(t) = 4.9t2] (green).
Fig. 5. Fb(t) graphs for a bungee jump (red: experimental and green: theoretical).
bungee jump. Predicted and measured values agree
very well.
Acknowledgments
I wish to thank my colleagues René Peerdeman and
Joost Osterhaus for making the figures and reading
the text, and also an anonymous referee for his/her
useful and encouraging remarks.
Appendix
Fig. 4. y(t) graphs for a bungee jump (red: experimental
and green: theoretical).
sensor.4 Figure 5 shows the predicted Fb(t) graph
(green) and the measured Fb(t) graph (red). Again,
the curves are seen to be in good agreement.
Summary
I have described an exercise for determining both
theoretically and experimentally the displacement y(t)
of a simulated bungee jumper. The apparatus and
methods are appropriate for use in an introductorylevel physics course. The modeling in this article is
performed using one of the modules of Coach, but
other spreadsheets or a BASIC program would also
work. Any interface that can handle voltages between
0 and 5 V can be used to feed the data into a computer. The advantage of Coach is that it allows the possibility of displaying the predicted graph and the measured graph on the same axes. With some modifications of the bungee system, the measurements could
be made using a motion sensor. I’ve also shown that it
is possible to investigate forces that act during a
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The y-ssensor
My y-sensor is cheap and took less than one hour to
make. It is, in fact, a potentiometer whose sliding element is the cylindrical brass “jumper” (4.0 cm in
length and 1.5 cm in diameter). I drilled an axial hole
(6 mm in diameter) into the bottom of the cylinder
extending almost to the top and lined the hole with a
piece of plastic tubing. I drilled a smaller (1.0 mm diameter) axial hole in the top of the cylinder and
passed a constantan wire (0.4 mm diameter) through
this hole and the rest of the cylinder. The constantan
wire is stretched vertically between two fixed points
(see Fig. 6). A stabilized voltage of 5.00 V is applied
across the wire and a small bolt is attached to the top
of the cylinder. A very thin copper wire is connected
to this bolt and lightly wrapped twice around the constantan wire. It’s critical that this wrapping be done
carefully — not overly tight (too much friction during
the fall) and not too loose (poor electrical contact).
To repress possible pulses caused by erratic contact, I
connect an 0.5-F capacitor across the input of the
computer. As the cylinder falls, the potential difference between the sliding contact and ground can be
THE PHYSICS TEACHER ◆ Vol. 41, April 2003
Fig. 6. Schematic diagram of the y-sensor.
etcetera...
fed into any interface that can handle voltages between 0 and 5 V. I used CMA’s CoachLab 2 in combination with Coach, which gave me the possibility of
generating theoretical and experimental graphs in one
integrated experiment. It would also be possible to
use the Vernier differential voltage probe in combination with LabPro.
References
1. D. Kagan and A. Kott, “The greater-than-g acceleration
of a bungee jumper,” Phys. Teach. 34, 368–373 (Sept.
1996).
2. Coach is a versatile activity-based environment that
makes it possible to: (a) collect data with all kinds of
sensors; (b) collect data from video-clips; (c) process
these data with advanced mathematical tools; and (d)
compare the experimental results with predictions obtained by modeling. Coach also supports Texas Instruments CBLs and Vernier LabPro. Coach is a product of
CMA, AMSTEL Institute, Universiteit van Amsterdam. See http://www.cma.science.uva.nl/english, or
http://www.harris-educational.com/Probeware.
3. M.G. Calkin and R.H. March, “The dynamics of a
falling chain: I,” Am. J. Phys. 57, 154–157 (1989).
4. The CMA Dual-Range Force Sensor B0362bt. The
Vernier Dual-Range Force Sensor DFS-BTA could also
be used.
PACS codes: 02.60, 06.90, 46.20B, 46.05B
Hubert Biezeveld has an M.S. degree in physics from the
University of Amsterdam. He has been teaching high
school physics since 1968 and co-authored a high school
physics textbook Scoop (Dutch word for Scope!). He also
was the editor of Faraday, the journal for Dutch teachers
in chemistry and physics, and received the Minnaert
Award for his didactic activities.
OSG West-Friesland, Hoorn, The Netherlands;
hubert.biezeveld@planet.nl
etcetera... Editor
Albert A. Bartlett, Department of Physics,
University of Colorado, Boulder, CO 80309-0390
Multiple-Choice Question1
Which of the following is not an answer to this question?
a) All of the responses below
b) This one
c) Some of the above
d) All of the above
e) None of these
1. Donald E. Simanek, The Vector 20, 29 (Fall 1991).
THE PHYSICS TEACHER ◆ Vol. 41, April 2003
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